Earnshaw's theorem states that it is impossible to have stable levitation in a magneto-static field. Although the theorem has a very broad scope, it requires some very precise conditions. In particular, the theorem fails if there are diamagnetic materials present, the levitating body is spinning, or the fields are alternating in time.
The discovery of the Levitron™ showed that it is possible to achieve stable levitation of a top in a steady magneto-static field. The Levitron™ is described in U.S. Pat. No. 5,404,062, incorporated herein by reference. Stable levitation has also been demonstrated using diamagnetic materials, one particular instance of which is levitation over a superconducting disc, which can be considered as a diamagnetic material with magnetic permeability of zero.
Previous work has shown that small particles can be levitated in an alternating magnetic field. This work has been outlined by W. Paul in Electromagnetic Traps for Charged and Neutral Particles, Rev. Modern Physics, 62 (1990), pp. 531-540 (hereinafter “Paul”), incorporated herein by reference. This has been applied beneficially in fields such as mass spectrometry, but the small particles are essentially point charges in the alternating electro-magnetic field, so rigid body dynamics of the levitated object can be neglected.
It is desirable in view of the foregoing to provide for levitating at a stable equilibrium, in an alternating magnetic filed, an object whose rigid body dynamics affect the levitation operation.
Some exemplary embodiments of the invention can achieve stable levitation of such an object by eliminating coupling between the rotational and translational forces acting on the object. Other exemplary embodiments of the invention can achieve stable levitation of such an object by varying the coupling between the rotational and translational forces acting on the object, while maintaining one or more of the rotational and translational forces steady in time.